Ph.D. Thesis - Physics

Simulation of a truncated harmonic oscillator
Here we explore the basic features of the truncated oscillator quantum simulation experi-
ment [STH + 98], as a guide to understanding some of the issues in NMR quantum simulation.
The Hamiltonian of the quantum harmonic oscillator is
HHO = Ω (N + 1/2) , (2.24)
where N = a † a is the operator for number of vibrational quanta. The solution is an infinite
“ladder” of energy eigenstates |n〉, with energy EHO = Ω (n + 1/2) and n ranging from 0
to ∞. The Hilbert space of the set of nuclei in NMR has a tensor product structure. In the
ˆz basis, the eigenstates that span the Hilbert space are {|↑↑〉,|↑↓〉,|↓↑〉,|↓↓〉}, where the
arrows represent the spin state of nuclei 1 and 2. The first step is to find a mapping from
this structure to the oscillator eigenstates. They make the following unitary mapping:
|n = 0〉 ↦→ |↑↑〉 ; (2.25)
|n = 1〉 ↦→ |↑↓〉 ; (2.26)
|n = 2〉 ↦→ |↓↓〉 ; (2.27)
|n = 3〉 ↦→ |↓↑〉 . (2.28)
Since there are four available basis states when using two qubits, the simulation will neces-
sarily be of a truncated harmonic oscillator. Higher levels could be simulated by additional
qubits; in fact, the number of oscillator levels doubles for each additional qubit. However,
doing this in NMR depends on the couplings between all individual qubits being strong
enough.
The Hamiltonian, mapped to the NMR system, has the following form:
Hsim = ωHO
1 3 5 7
|↑〉 〈↑| + |↑〉 〈↓| + |↓〉 〈↑| + |↓〉 〈↑| . (2.29)
2 2 2 2
This Hamiltonian is implemented by using appropriate refocusing pulses (Sec. 2.1.5).
The state preparation is done using logical labeling, rather than the temporal labeling
covered previously. Suppose we begin with a superposition state such as |Ψi〉 = |↑↑〉+i |↓↓〉.
What would we then expect to observe from the above quantum circuit? Eigenstates of the
Hamiltonian given by Eq. 2.29 will not evolve, but we do expect a quantum superposition
state to evolve at a rate corresponding to the energy difference between the states that form
it.
The experiment was done using the two hydrogen nuclei in the 2,3-dibromothiophene
molecule (Fig. 2-2). Each of these nuclei is a single proton, and the proton Larmor frequency
in their spectrometer is 400 MHz. Other pertinent quantities are (ω2 −ω1)/(2π) = 226 MHz
and J = 5.7 Hz. The experimental results are shown in Fig. 2-3. The simulation is a
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